An upper bound for the minimum weight of the dual codes of desarguesian planes

Abstract

Abstract We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529–538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p m where p is a prime, and m ≥ 1 . This gives words of weight 2 p m + 1 − p m − 1 p − 1 in the dual of the p -ary code of the desarguesian plane of order p m , and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [Gábor Korchmáros, Francesco Mazzocca, On ( q + t ) -arcs of type ( 0 , 2 , t ) in a desarguesian plane of order q , Math. Proc. Cambridge Philos. Soc. 108 (1990) 445–459].

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